/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Tabulate and plot enough points ... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 37

Tabulate and plot enough points to sketch a graph of the following equations. $$r=8 \cos \theta$$

Short Answer

Expert verified
The points obtained for the values of \(\theta\) are as follows: | \(\theta\) | \(r\) | |---------|-----| | 0 | 8 | | \(\frac{\pi}{4}\) | \(4\sqrt{2}\) | | \(\frac{\pi}{2}\) | 0 | | \(\frac{3\pi}{4}\) | \(-4\sqrt{2}\) | | \(\pi\) | -8 | | \(\frac{5\pi}{4}\) | \(-4\sqrt{2}\) | | \(\frac{3\pi}{2}\) | 0 | | \(\frac{7\pi}{4}\) | \(4\sqrt{2}\) | | \(2\pi\) | 8 | The shape of the graph is a circle centered at \((4,0)\) with a radius of 4.

Step by step solution

01

Analyze the Equation

Analyze the given equation: \(r=8\cos\theta\). This is a polar coordinate equation where \(r\) is the distance of each point from the origin and \(\theta\) is the angle each point makes with the positive x-axis.
02

Choose Values for \(\theta\) and Calculate \(r\)

Choose a few values for \(\theta\) to calculate \(r\) for different points. We can use angles from 0 to \(2\pi\). Remember to work in radians. For this example, we can choose the following values for \(\theta\): $$\theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}, 2\pi$$ Now, calculate the corresponding value of \(r\) for each angle: \(r = 8\cos(\theta)\)
03

Tabulate the Data

Create a data table to organize the values of \(\theta\) and their corresponding value of \(r\): | \(\theta\) | \(r\) | |---------|-----| | 0 | 8 | | \(\frac{\pi}{4}\) | \(4\sqrt{2}\) | | \(\frac{\pi}{2}\) | 0 | | \(\frac{3\pi}{4}\) | \(-4\sqrt{2}\) | | \(\pi\) | -8 | | \(\frac{5\pi}{4}\) | \(-4\sqrt{2}\) | | \(\frac{3\pi}{2}\) | 0 | | \(\frac{7\pi}{4}\) | \(4\sqrt{2}\) | | \(2\pi\) | 8 |
04

Sketch the Graph

Plot the points from the table on a polar-coordinate grid using polar axis or by converting them to Cartesian coordinates if needed. After plotting the points and connecting them, the sketch of the graph should look like a circle centered at \((4,0)\) with a radius of 4.

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Most popular questions from this chapter

A simplified model assumes that the orbits of Earth and Mars are circular with radii of 2 and \(3,\) respectively, and that Earth completes one orbit in one year while Mars takes two years. The position of Mars as seen from Earth is given by the parametric equations $$x=(3-4 \cos \pi t) \cos \pi t+2, \quad y=(3-4 \cos \pi t) \sin \pi t$$ a. Graph the parametric equations, for \(0 \leq t \leq 2\) b. Letting \(r=(3-4 \cos \pi t),\) explain why the path of Mars as seen from Earth is a lima莽on.

Completed in 1937, San Francisco's Golden Gate Bridge is \(2.7 \mathrm{km}\) long and weighs about 890,000 tons. The length of the span between the two central towers is \(1280 \mathrm{m} ;\) the towers themselves extend \(152 \mathrm{m}\) above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway \(500 \mathrm{m}\) from the center of the bridge?

Show that the equation \(r=a \cos \theta+b \sin \theta\) where \(a\) and \(b\) are real numbers, describes a circle. Find the center and radius of the circle.

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{12}{3-\cos \theta}$$

Find the area of the regions bounded by the following curves. The lima莽on \(r=4-2 \cos \theta\)
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